thermodynamic optimization of global circulation and climate

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2005; 29:303–316Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/er.1058

Thermodynamic optimization of global circulation and climate

Adrian Bejan1,n,y and A. Heitor Reis2

1Department of Mechanical Engineering and Materials Science, Duke University, Box 90300,

Durham, NC 27708-0300, U.S.A.2Department of Physics, University of !EEvora, Colegio Luis Verney, Rua Rom *aao Ramalho, 59,

7000-671 !EEvora, Portugal

SUMMARY

The constructal law of generation of flow structure is used to predict the main features of global circulationand climate. The flow structure is the atmospheric and oceanic circulation. This feature is modelled asconvection loops, and added to the earth model as a heat engine heated by the Sun and cooled by thebackground. It is shown that the dissipation of the power produced by the earth engine can be maximizedby selecting the proper balance between the hot and cold zones of the Earth, and by optimizing the thermalconductance of the circulation loops. The optimized features agree with the main characteristics of globalcirculation and climate. The robustness of these predictions, and the place of the constructal law as a self-standing principle in thermodynamics, are discussed. Copyright # 2005 John Wiley & Sons, Ltd.

KEY WORDS: constructal theory; climate; global circulation; thermodynamic optimization

1. INTRODUCTION

The purpose of this paper is to establish the connection between two domains: the climatemodels that have been developed ad hoc in geophysics, and the heat engine models that havebeen developed ad hoc for wind power calculations in thermodynamics. Both domains are veryactive, voluminous, and established}to review them is not the objective. The connection is madeon the basis of constructal theory (Bejan, 1997, 2000; Bejan et al., 2004), which is the thoughtthat thermodynamic systems far from equilibrium (flow systems) construct flow architectureswith purpose: the maximization of global system performance subject to global constraints. Thepurpose of making the connection is to eliminate the ‘ad hoc’ from both fields, and to show that asingle principle rules all the model optimizations that have been demonstrated so far.

This connection is important because atmospheric and oceanic circulation is the largest flowsystem on Earth, and because the implications of the theoretical success demonstrated in the twodomains are great. In geophysics, the invocation of a principle of maximization of entropygeneration rate has been recognized as a ‘law of maximum entropy production which followsdeductively from the second law of thermodynamics’ (Swenson, 1989). If that were the case, the

Received 16 July 2003Accepted 31 March 2004Copyright # 2005 John Wiley & Sons, Ltd.

nCorrespondence to: Adrian Bejan, Department of Mechanical Engineering and Materials Science, Duke University,Box 90300, Durham, NC 27708-0300, U.S.A.yE-mail: [email protected]

principle invoked in geophysics is at best a theorem, not a new law. More recent work onconstructal theory (Bejan, 2000) suggests that the principle is indeed a self-standing law. Theemergence of globally optimized flow architectures in all classes of flow systems (animate,inanimate, engineered) is a universal phenomenon (Bejan, 2000).

Climate models based on the maximization of entropy generation rate began with Malkus’(1954) hypothesis that in B!eenard convection the overall heat transfer rate is maximized throughthe selection of flow pattern. Examples and reviews are available in (Lorenz, 1955; Paltridge,1975, 1978; Schulman, 1977; North, 1981; Lin, 1982; Lorenz et al., 2001). The connectionbetween this hypothesis and constructal theory was noted in Nelson and Bejan (1998), where theconstructal principle delivered all the known characteristics of B!eenard flow and heat transfer, influids as well as fluid-saturated porous media. In this paper, constructal theory is extended to theproblem of atmospheric and oceanic circulation driven by heating from the sun.

The method consists of viewing the Sun–Earth–Universe assembly as an extraterrestrialpower plant the power output of which is used for the purpose of forcing the atmosphere andhydrosphere to flow. The power plant models that have been proposed and optimized are listedchronologically in (Bejan, 1988; Gordon and Zarmi, 1989; De Vos, 1992; De Vos and Flater,1991; De Vos and Van der Wel, 1993). A spaceship, or the Earth, may be viewed as a closedsystem having two surfaces, a hot surface of area AH and temperature TH; which is heated by theSun, and a cold surface ðAL;TLÞ cooled by radiation to the Universe. On a spaceship, thecollector ðAHÞ and radiator ðALÞ are the object of design (Bejan, 1988). In the modelling of windgeneration on Earth, the surfaces AH and AL represent the daily illuminated and darkhemispheres (Gordon and Zarmi, 1989; De Vos, 1992; De Vos and Flater, 1991), or the time-averaged equatorial and polar zones (De Vos, 1992; De Vos and Van der Wel, 1993). In allcases, the total radiation heat transfer surface is fixed,

AH þ AL ¼ A ð1Þ

Global constraints of this type play a central role in constructal theory. In all the power plantmodels (Bejan, 1988; Gordon and Zarmi, 1989; De Vos, 1992; De Vos and Flater, 1991; De Vosand Van der Wel, 1993) it was found that the power output can be maximized by selecting twoparameters, the area fraction ðAH=AÞ; and the position of the power plant on thethermodynamic temperature scale, for example, the temperature TH:

Thermodynamic optimization authors (Bejan, 1988; Gordon and Zarmi, 1989; De Vos, 1992;De Vos and Flater, 1991; De Vos and Van der Wel, 1993) are unaware of the use of similarprinciples in geophysics. At the same time, geophysics authors (Lorenz et al., 2001) continue tobe unaware of the thermodynamics literature. To bring these two research worlds together isone of the objectives of this paper. To begin with, none of the power plant models dedicated tothe generation of wind on Earth (Bejan, 1988; Gordon and Zarmi, 1989; De Vos, 1992; De Vosand Flater, 1991; De Vos and Van der Wel, 1993) have accounted for natural convection}themechanism by which the generated power is dissipated into heat by the movement of theatmosphere and the oceans. This mechanism is the new feature that the present paper adds tothe power plant models that have been proposed. The mechanism has been recognized before,not only in the modelling of atmospheric dynamics (Lin, 1982; Lorenz et al., 2001), but also inthe description of natural convection itself. See, for example, p. 111 in (Bejan, 1984). . .theconvection loop is equivalent to the cycle executed by the working fluid in a heat engine. Inprinciple, this heat engine cycle should be capable of delivering useful work if we insert

Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Energy Res. 2005; 29:303–316

A. BEJAN AND A. H. REIS304

a propeller in the stream: this is the origin of ‘wind power’ discussed nowadays in connectionwith the harnessing of solar work indirectly from the atmospheric heat engine loop. In theabsence of work-collecting devices, the heat engine cycle drives its working fluid fast enough sothat its entire work output potential is dissipated by friction in the brake at the interface betweenwhat moves and what does not move’.

2. EARTH MODEL WITH NATURAL CONVECTION LOOPS

The main features of the model are illustrated in Figures 1 and 2. The surface temperature istime independent. It is averaged over the daily and annual cycles, and is represented by two

Figure 1. Earth model with equatorial (AH) and polar (AL) surfaces, andconvective heat current between them.


THERMODYNAMIC OPTIMIZATION 305

temperatures ðTH;TLÞ that correspond to the equatorial and polar zones ðAH;ALÞ: This surfacemodel and radiation heat transfer calculations are the same as in De Vos (1992) and De Vos andVan der Wel (1993). The equatorial surface receives the solar heat current

qs ¼ AHp 1� rð ÞfsT4s ð2Þ

where Ts, s, f and r are the temperature of the Sun as a black body (5762K), the Stefan–Boltzmann constant (5.67� 10�8Wm�2K�4), the Earth–Sun view factor (2.16� 10�5), andthe albedo of the Earth (0.3). The area AHp is the area AH projected on a plane perpendicular tothe direction Earth–Sun. The ratio AHp/AH decreases from 1/p when AH is a narrow belt alongthe equator, to 1

4when AH covers the globe almost completely. For simplicity, we adopt the

approximation AHp/AH = 14; which is independent of the variable area fraction x ¼ AH=A: This

approximation is permissible in view of the order-of-magnitude analysis of the naturalconvection part of the model.

The AH surface radiates into space the heat current

q1 ¼ AH 1� gð ÞsT4H ð3Þ

where g ¼ 0:4 is the Earth’s greenhouse factor, or the reflectance in the infrared region (De Vosand Flater, 1991). The difference between qs and q1 is convected over the Earth’s surface, fromAH to AL,

q ¼ qs � q1 ð4Þ

and it is finally transmitted by radiation to the cold background,

q ¼ AL 1� gð ÞsT4L ð5Þ

Equations (3) and (5) have been simplified by neglecting T41 in favour of T4

L and T4H; where

T1 = 3K is the temperature of the background, and by considering the Earth emissivity equalto unity. By combining Equations (1)–(5) with the notation x ¼ AH=A; we obtain

T4H þ

1

x� 1

� �T4L ¼ B ð6Þ

Figure 2. Natural convection loops in a fluid layer connecting the equatorial and polar surfaces.



where B is nearly constant,

B ¼f

4T4s

1� r1� g

ffi 7� 109 K4 ð7Þ

The heat current q is driven from TH to TL by the buoyancy effect in the layer of fluid thatcovers the Earth’s surface. In the following derivation of the q formula we neglect factors oforder 1, in accordance with the rules of scale analysis (Bejan, 1984). The fluid layer covers anarea of flow length Lð� RÞ and width Wð� RÞ; where R is the Earth’s radius. The verticallength scale of the fluid layer, H, will be defined shortly. The length L bridges the gap betweenTH and TL.

At the TH-end of the fluid layer, the hydrostatic pressure at the bottom of the layer is rHgH.Similarly, at the TL-end the pressure is rLgH. The pressure difference in the L direction is

DP � rL � rH� �

gH � rb TH � TLð ÞgH ð8Þ

where r is the mean fluid density, and b is the coefficient of volumetric thermal expansion.The fluid-layer control volume is exposed to the force DPWH in the L direction. This force is

opposed by the shear force felt by the moving fluid over the surface LW,

DPWH ¼ tLW ð9Þ

The average shear stress is

t � reMu

Hð10Þ

where eM is the eddy diffusivity for momentum, and u is the velocity in the L direction. For theorder of magnitude of eM we use Prandtl’s mixing length model (e.g., Bejan, 1984, p. 236), inwhich we take H to represent the mixing length,

eM ¼ H2 u

H¼ Hu ð11Þ

In other words, H is the vertical dimension of the fluid system that mixes (transfers momentumvertically) while moving horizontally. Note that H is not the vertical extent of the fluid layer. Byeliminating DP; u and eM between Equations (8)–(11) we obtain the horizontal velocity scale

u � bg TH � TLð ÞH2

L

� �1=2ð12Þ

The convective heat transfer rate associated with the counterflow between TH and TL dependson whether the two branches of the counterflow are in intimate thermal contact. They are not if,for example, the circulation is in the plane L�W ; as in the case of R-scale oceanic andatmospheric currents that complete loops over large portions of the globe (Figure 2(a)). Anotherexample is when the loop is a vertical plane aligned with the meridian, when the branches of thecounterflow are far enough apart and do not exchange heat in a significant way in the verticaldirection (Figure 2(b)). In such cases the convective heat current is

q � ruHWcp TH � TLð Þ ð13Þ

or, after using Equation (12) and L � W � R;

q � rcp gbð Þ1=2H2R1=2 TH � TLð Þ3=2 ð14Þ



The convective current is proportional to (TH –TL)3/2, not to (TH –TL), as it might have been

assumed based on a simple invocation of convection as a mechanism (e.g., Equation (15)below). To verify that Equation (14) provides a realistic estimate of the q scale, note that in thegeophysics literature (North, 1981) the group

D �q

R2 TH � TLð Þð15Þ

is known as the convective conductance in the horizontal direction, expressed per unit ofhorizontal (earth) area. Note that the proportionality between heat current and temperaturedifference in the D definition is assumed, not deduced (we return to this aspect in Section 4 andTable I). The value of D is known empirically (Lorenz et al., 2001) to be of the order of0.6Wm�2K�1. The theoretical estimate (14) anticipates

D � rcp gbð Þ1=2H2R�3=2 TH � THð Þ1=2 ð16Þ

which for R � 5000 km, (TH–TL) � 40K, and air properties evaluated at 08C yields

D � 0:14H2 km�2 Wm�2 K�1 ð17Þ

This estimate agrees with the empirical D value when H is of the order of 2 km, which is arealistic transversal length scale for flows in the atmosphere. If we perform the correspondingcalculation using the properties of water at 108C, then in Equation (17) the coefficient 0.14 isreplaced by 70, and D matches 0.6Wm�2K�1 when H is of order 100m: this length scale iscompatible with the thickness of mixing layers in the ocean.

Consider now the alternative where the two branches of the counterflow are in good thermalcontact. If the counterflow is a loop in the vertical plane L�H; with the warm branch flowingfrom TH to TL over the top, and the cold branch returning along the bottom (Figure 2(b)), thenEquation (12) continues to hold, but Equation (13) is replaced by

q � ruHWcpDT ð18Þ

where DT is the top–bottom temperature difference between the two branches of thecounterflow, DT5TH2TL: The temperature of the warm branch decreases from TH at thewarm entrance, to TL þ DT at the cold exit. The enthalpy drop in the horizontal direction iscaused by the heat transfer by eddy motion in the vertical direction,

uH TH � TL � DTð Þ � eHLDTH

ð19Þ

Table I. The effect of the heat transfer model on the results of the double maximization of the powerproduced and destroyed by global circulation.

n wmm (W) xopt Cn,opt TL (K) TH (K)

1 2.28 � 108 0.35 1.2 � 107K3 189.9 260.23/2 2.28 � 108 0.35 1.4 � 106K5/2 189.5 260.52 2.28 � 108 0.34 1.6 � 105K2 188.1 260.0



where eH is the thermal eddy diffusivity. Using again the mixing length model, eH � uH; we findthat Equation (19) reduces to

DTTH � TL

�H

L{1 ð20Þ

such that the convection current (18) becomes

q � rcp gbð Þ1=2H3R�1=2 TH � TLð Þ3=2 ð21Þ

D � rcp gbð Þ1=2H3R�5=2 TH � TLð Þ1=2 ð22Þ

These estimates for q and D are less realistic because they are smaller by a factor of H=R{1than those of Equations (14) and (16).

In conclusion, the convection model with counterflow branches that are not in intimatethermal contact is more realistic. Such a counterflow poses a minimal convective resistance toheat flow in the flow direction. This feature has been encountered before. The same type ofcounterflow, i.e. the same minimal resistance principle was used earlier in constructal theory inthe prediction of B!eenard convection (Nelson and Bejan, 1998) and the optimization of cavitiesin walls with many horizontal partitions (Bejan, 1980) (see also Bejan, 2000).

In the following analysis, we retain Equation (14) because of the test given under Equation(17). Combining Equations (14) and (5), AL ¼ ð1� xÞA and A ¼ 4pR2; we arrive at

C3=2 TH � TLð Þ3=2� 1� xð ÞT4L ð23Þ

where the theoretical conductance is

C3=2 ¼rcp gbð Þ1=2H2R1=2

4pR2 1� gð Þs� 2:1� 105 K5=2 ð24Þ

This C3=2 value corresponds to air at 1 atm and 08C, and the assumption that H � 2 km.In summary, the radiation–convection model reduces to Equations (6) and (23), which

determine the functions THðxÞ and TLðxÞ: Unlike in earlier thermodynamic optimizations(Gordon and Zarmi, 1989; De Vos, 1992; De Vos and Flater, 1991; De Vos and Van der Wel,1993), the only degree of freedom in this configuration is the area fraction x. Figure 3 shows theeffect of x on TH and TL. In the limit x = 1, or AH = A, the Earth’s surface has a uniformtemperature, 289K. The constants B and C3=2 are known, although the C3=2 estimate is notnearly as accurate as B. We may treat the convective conductance C3=2 as a parameter thataccounts for the accuracy of the convection model (more on this in Table I).

The fact that the collector–radiator temperature difference ðTH2TLÞ participates in the modelas ðTH2TLÞ

3=2 is in agreement with the Monin–Obukhov similarity theory (Pal Arya, 1988),provided that the vertical length scale H is replaced by the Obukhov length,

LO ¼u3rcpkgbq

ð25Þ

where k � 0:24 is von Karman’s constant. The assumption H � LO is reasonable because theObukhov length is the scale where the atmospheric shear effects are dominant.



In brief, according to Monin–Obukhov theory, the mean flow and turbulent characteristicsof the Earth boundary layer depend on only four independent parameters: (i) the height z abovethe earth surface, (ii) the friction velocity un ¼ ðt=rÞ1=2 cf. (Pal Arya, 1988, p. 144), where un isthe same as u of Equations (10) and (11), (iii) the surface kinematic heat flux q00=ðrcpÞ; and (iv) thebuoyancy effect gb; or g=T (cf. Pal Arya, 1988, p. 157). Buckingham’s p theorem states that theonly dimensionless combination of these four variables consists of z=LO and the LO group shownin Equation (25). In the analysis presented in this section we arrived at the same conclusion in adifferent way. The key assumption was made in Equation (9), where the effective vertical lengthscale (H) was taken to be the same as the thickness of the zone dominated by shear.

3. MAXIMIZATION OF MECHANICAL POWER DISSIPATION

The earth surface model with natural convection loops allows us to estimate several quantitiesthat characterize the global performance of atmospheric and oceanic circulation. We pursue thisfrom the constructal point of view, which is that the circulation itself represents a flow geometrythat is the result of the maximization of global performance subject to global constraints.

The first quantity is the mechanical power that could be generated by a power plant operatingbetween TH and TL, and driven by the heat input q. The power output (w) is dissipated byfriction in fluid flow (a fluid brake system), and added fully to the heat current (qL) that thepower plant rejects to TL. The power plant and the fluid brake system occupy the space shownbetween AH and AL in the lower part of Figure 1. Note the continuity of the heat current qthrough this composite system, q ¼ wþ qL: The most power that the composite system couldproduce is associated with the reversible operation of the power plant. The power output in this

Figure 3. The effect of the area allocation fraction x on the collector and radiator temperatures.



limit is proportional to

w ¼ q 1�TL

TH

� �ð26Þ

which is shown as a function of x in Figure 4. The w maximum is due to two competing trends.When x approaches 1, q tends to zero (see Equation (5)), the Earth’s surface becomes isothermalðTL=TH ¼ 1Þ; and the power output w is zero. In the opposite limit (x= 0), TL tends to zero (seeEquation (6)), and w tends to zero because q vanishes. The maximum power (wm) occurs whenxopt = 0.35. This point of optimal thermodynamic operation (xopt, wm, TH, TL) is reported inFigures 5–7, where the abscissa parameter is the convective conductance C3=2; which is allowedto vary.

The maximized power (wm) exhibits a maximum with respect to the convective conduc-tance, as shown in Figure 6. The maximum is located at C3=2;opt¼ 1:4� 106 K5=2 and wmm¼2:28� 108 W: Note that the optimal convective conductance C3/2,opt exceeds by a factor ofonly 6 the value anticipated in Equation (24). In view of the approximate character of the simplemodel (radiation, Earth surface, convection) employed in this analysis (Figures 1 and 2), weconclude that the agreement between C3/2,opt and Equation (24) is correct in an order ofmagnitude sense. On the same basis, we speculate that the natural counterflow structuredescribed in Figure 2 is the result of the tendency of the flow system to dissipate maximumpower, such that, if suddenly left alone (in isolation), the flow system would reach equilibriumthe fastest. This tendency is evident in all flows, and is another way of stating the constructal law(cf. Bejan, 2000, Chapters 5 and 6).

It is also possible to regard the agreement between Equation (24) and C3/2,opt as a coincidence.The reason for claiming more than a coincidence is the observation that similar coincidences are

Figure 4. The maximization of power output, convective heat transfer rate, and convection thermalconductance, with respect to the surface allocation ratio.



Figure 5. The optimal area allocation ratio resulting from the optimization shown in Figure 4.

Figure 6. The maximized global performance indicator resulting from theoptimizations shown in Figure 4.



occurring every time the global performance maximization principle is invoked. Examplesmost relevant to the present are the prediction of the flow structure and global Nusselt numberfor B!eenard convection (Nelson and Bejan, 1998) and the laminar–turbulent transition (orturbulent–laminar) in many flow configurations (Bejan, 2000). The present example adds itselfto the list, and suggests that global circulation and climate are covered (anticipated) by theconstructal law.

4. THE EFFECT OF EVAPORATION/CONDENSATION OF WATERIN THE ATMOSPHERE

Better agreement between the present theory and observed properties of atmospheric circulationis obtained if we account for the effect of latent heat in heating and cooling of humid air. In theanalysis of Section 2, we used only the sensible heat effect. The ratio between the sensible heat andlatent heat effects in the atmosphere is known as the Bowen ratio (Bo). According to Brutsaert(1991), several Bo estimates are available and are of the same order of magnitude, for example:Budyko (Bo=5), Reichel (Bo=3.8), and Korzum et al. (Bo=5.6) (cf. (Brutsaert, 1991) p. 5). TheBo effect can be added to the preceding model by dividing the right side of Equation (14) by Bo.

5. CONCLUSIONS

Earlier applications of the constructal principle (Bejan, 2000) recommended it as a self-standinglaw that is distinct from the second law. There is nothing in classical thermodynamics to requirethe Carnot power to exhibit maxima with respect to geometric features such as x and C3=2: That

Figure 7. The temperatures of the two surfaces at the optimum summarized in Figures 5 and 6.



the optimized geometric features are consistent with features observed in nature affirms the lawstatus of the principle. It is a thermodynamic principle because it completes the thermodynamicdescription of flow systems under constraints. Flow and dissipation (irreversibility) must bedistributed in special ways in space and time.

Figures 4–6 showed two alternatives to the maximization of power production anddissipation. One is the convective heat current (q) between the two temperature zones. Theother is the ratio q=TH; which, in view of the assumption T1{TH; is essentially the same asthe thermal conductance q=ðTH2T1Þ: These two quantities can be maximized with respect tothe partitioning of the Earth’s surface (x). Figure 5 shows that the optimal area allocation ratiois almost the same as when w is maximized.

The difference is displayed in Figure 6: qm and (q/TH)m do not exhibit maxima with respect tothe convective conductance C3=2: This makes the maximization and subsequent dissipation ofpower the more likely objective from the constructal point of view. Why? Because theconstructal principle calls for the generation of flow architecture, and architectural features suchas xopt and C3/2,opt are unambiguously a part of nature.

The robustness exhibited by xopt with respect to the optimization objective (Figure 5) isintriguing. We may ask whether the natural circulation model has any effect on xopt. Forexample, according to the simplest heat transfer model of convection, the heat current betweenthe warm and cold regions of the earth would be proportional to the temperature difference(TH�TL), not to (TH�TL)

3/2 as in Equations (14) and (21). This would replace Equation (23)with the more general statement

Cn TH � TLð Þn� 1� xð ÞT4L ð27Þ

where n ¼ 1 represents the simplest model, and n ¼ 32the convection model used in Section 3.

The units of Cn are K4�n: Table I shows the results of repeating for n ¼ 1 and 2 the optimization

work described in Section 3. The area allocation ratio and the maximized power are insensitiveto the heat transfer model in the n range that can be considered reasonable. The optimizedthermal conductance factor Cn;opt depends on the heat transfer model. The important feature ofCn is that it can be optimized, i.e. from an infinity of possible Cn values, there is one for whichthe global generation and dissipation of power via organized flow is maximal.

To see why the constructal law of optimization of flow architecture is distinct from the secondlaw, let us review the essence of thermodynamics. Thermodynamic theory was developed inorder to account for the functioning and improvement of heat engines. The first law accountsfor the conservation of energy, and serves as definition of energy as a concept. The second lawaccounts for the generation of entropy (irreversibility), on the one-way nature of currents thatovercome resistances, and serves as definition of entropy as a concept.

The first law and the second law account for the functioning of a given (observed, assumed)heat engine configuration. The history of flow systems (e.g. heat engines) shows what the firstlaw and the second law are not covering: the case-by-case increase in performance, in time.Every class of flow systems exhibits this behaviour, from river basins to animals and heatengines. Each system in its class represents a flow architecture. New flow systems coexistwith old systems, but persist in time if they are better, while older systems gradually disappear.This never-ending parade of flow systems represents the generation of flow architecture}thegeneration of geometric form as the clash between objective and constraints in flow systems.This is the phenomenon summarized in the constructal law.



NOMENCLATURE

AH =high-temperature area (m2)AHp = projected high-temperature area (m2)AL = low-temperature area (m2)B =constant (K4), Equation (7)Bo =Bowen ratiocp =specific heat at constant pressure (J kg�1K�1)C3/2 = thermal conductance (K5/2)Cn =general conductance (K4–n)D =convective conductance (Wm�2K�1)F =Earth–Sun view factorG =gravitational acceleration (m s�2)H =height (m)k =von Karman’s constantL = length (m)LO =Obukhov length (m)q =convection current (W)qs = solar heat current (W)q00 =heat flux (Wm�2)R =earth radius (m)TH =high temperature (K)TL = low temperature (K)Ts = Sun temperature (K)u =horizontal velocity (m s�1)un = friction velocity (m s�1)w =power output (W)W =width (m)x =hot area fraction, AH/Az =altitude (m)

Greek letters

b =coefficient of thermal expansion (K�1)g =Earth greenhouse factorDP =pressure difference (Pa)DT = temperature difference (K)eH = thermal eddy diffusivity (m2 s�1)eM =momentum eddy diffusivity (m2 s�1)r =Earth albedorH =density of warm air (kgm�3)rL = density of cold air (kgm�3)s =Stefan–Boltzmann constant (Wm�3K�4)t =average shear stress (Pa)



Subscripts

H =high temperatureL = low temperaturem =maximizedmm =maximized twiceopt = optimize

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